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I am totally confused with multivariate differentiation. So, how will I find the pair $(x,y)$ making below function optimum?

$$3x^2 + 6x + 5y^2 + 5y$$

P.S.: This is not a homework.

Amzoti
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user5054
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1 Answers1

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Differentiate with respect to $x$, treating $y$ as constant: $$ 6 x + 6 = 0 $$ Do the same for $y$ $$ 10 y + 5 = 0 $$ So the answer is $x=-1$, $y=-0.5$

user44197
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  • OK, thanks, then how about if we had a term including both $x$ and $y$, which will cause the derivative with respect to $x$ to be dependent to $y$, and vice versa? Then how will we find the optimum? Is that the case where we need gradient descent or coordinate descent to find the optimum? Or can we get the result by solving the linear system of 2 equations of 2 variables which we are left after taking the derivatives wrt $x$ and $y$? – user5054 Jan 01 '14 at 03:08
  • Let's make up an example: $ \ f(x,y) \ = \ 3x^2 + 6x - 2xy + 5y^2 + 5y \ . $ The partial derivative with respect to $ \ x \ $ is $ \ 6x + 6 - 2y \ $ and that with repsect to $ \ y \ $ is $ \ 10y + 5 - 2x \ $ . Just as you suggest at the end, you would solve the pair of simultaneous equations $ \ 6x + 6 - 2y \ = \ 0 \ $ and $ \ 10y + 5 - 2x \ = \ 0 \ . $ Happy New Year! – colormegone Jan 01 '14 at 03:43
  • But sometimes optimization function is of degree 3 or more then following above process you will get 2 or 3 pairs of such points, now to decide which one is optimal ;calculate hessian matrix and see is it positive or negative definate at particular that point. – Sry Jan 01 '14 at 04:24
  • one more thing it is not necessary that points given by taking partial derivative 0 need not be optimal always e.g. in ur posted question if you have to max. the function then (-1,-0.5) is not the answer. In fact function has no maxima. – Sry Jan 01 '14 at 04:28
  • @Sry Based on the question, I am not sure OP has learnt about Hessians, saddles etc. – user44197 Jan 01 '14 at 08:22